Question

Suppose a coin having probability \(0.7\) of coming up heads is tossed three times. Let \(X\) denote the number of heads that appear in the three tosses. Determine the probability mass function of \(X\).

Short answer

The probability mass function of the random variable X is given by: \[P(X = 0) = 0.027\] \[P(X = 1) = 0.189\] \[P(X = 2) = 0.441\] \[P(X = 3) = 0.343\]

Step-by-step solution

Determine possible values of X

X can take values from 0 to 3, representing the number of heads obtained in the three coin tosses: {0, 1, 2, 3}

Apply binomial probability formula for each value of X

Now we can apply the binomial probability formula with n = 3 and p = 0.7 for each value of X:

X = 0

\[P(X = 0) = \binom{3}{0} (0.7)^0 (0.3)^{3-0}\] \[P(X = 0) = 1 * 1 * (0.3)^3\] \[P(X = 0) = 0.027\]

X = 1

\[P(X = 1) = \binom{3}{1} (0.7)^1 (0.3)^{3-1}\] \[P(X = 1) = 3 * 0.7 * (0.3)^2\] \[P(X = 1) = 0.189\]

X = 2

\[P(X = 2) = \binom{3}{2} (0.7)^2 (0.3)^{3-2}\] \[P(X = 2) = 3 * 0.49 * 0.3\] \[P(X = 2) = 0.441\]

X = 3

\[P(X = 3) = \binom{3}{3} (0.7)^3 (0.3)^{3-3}\] \[P(X = 3) = 1 * (0.7)^3 * 1\] \[P(X = 3) = 0.343\]

Summarize the probability mass function

Now that we have the probabilities for each value of X, we can write the probability mass function using these values: \[P(X = 0) = 0.027\] \[P(X = 1) = 0.189\] \[P(X = 2) = 0.441\] \[P(X = 3) = 0.343\] So, with the provided probabilities, we have successfully determined the probability mass function of X.

Key concepts

Binomial Probability

Binomial probability is a fundamental concept in probability theory that applies to scenarios where there are two possible outcomes for each trial, often labeled as 'success' and 'failure'. In our exercise, tossing a coin and getting heads is considered a success, while tails is a failure. The probability of success on any given trial is constant, and each trial is independent of the others.

The binomial probability formula is used to calculate the likelihood of getting exactly 'k' successes in 'n' independent trials. This formula is represented as:
\[P(X = k) = \binom{n}{k} \cdot p^k \cdot (1-p)^{n-k}\]
where \(\binom{n}{k}\) is the number of ways to choose 'k' successes from 'n' trials (combinatorics), 'p' is the probability of success on a single trial, and \((1-p)\) is the probability of failure.

For the exercise where we want to determine the number of heads when a coin is tossed three times, we would use the binomial probability formula to find the probability for 0, 1, 2, and 3 heads.

Discrete Random Variables

Discrete random variables are variables that can take on a finite or countable infinite number of different values. Each possible value a discrete random variable can take on has an associated probability. These variables are the cornerstone of statistical analysis and probability theory, especially when dealing with experiments or processes that have distinct outcomes.

In the context of our coin toss exercise, the discrete random variable \(X\) represents the number of heads obtained in the three coin tosses, which can be 0, 1, 2, or 3. Each of these values has a corresponding probability, making up the variable's probability mass function. Understanding discrete random variables is essential for grasping the significance of different outcomes in a probabilistic experiment.

Probability Theory

Probability theory is the branch of mathematics that deals with the analysis and interpretation of random phenomena. It provides the foundation for quantifying how likely events are to occur. This theory encompasses various principles and techniques that allow us to calculate probabilities in complex situations and make informed predictions.

Within this framework, the probability mass function (pmf) plays a crucial role. It defines the probability that a discrete random variable is exactly equal to some value. The concept is central to understanding binomial probability, as seen in the coin toss exercise, where we calculate the pmf for getting a certain number of heads. Probability theory is not just a purely academic study; it has practical applications in fields such as finance, risk assessment, and even everyday decision-making.

Combinatorics

Combinatorics is the area of mathematics that studies the counting, arrangement, and combination of objects. It is essential in solving problems related to probability, especially when we deal with discrete random variables and binomial probability.

One of the fundamental concepts in combinatorics is the combination, represented by \(\binom{n}{k}\), which calculates how many ways 'k' objects can be selected from a group of 'n' objects without regard to the order. In our example of a coin being tossed three times, this idea is used to determine the number of ways to achieve a certain number of heads. Using combinatorics, we can systematically calculate the likelihood of different outcomes and thus construct a probability mass function for a given scenario.